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In the above diagram, the 16 dots are in rows and columns, [#permalink]
03 Feb 2014, 10:47

Expert's post

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A

B

C

D

E

Difficulty:

95% (hard)

Question Stats:

32%(02:15) correct
68%(01:52) wrong based on 53 sessions

Attachment:

4x4 grid.JPG [ 9.78 KiB | Viewed 1372 times ]

In the above diagram, the 16 dots are in rows and columns, and are equally spaced in both the horizontal & vertical direction. How many triangles, of absolutely any shape, can be created from three dots in this diagram? Different orientations (reflections, rotations, etc.) and/or positions count as different triangles. (Notice that three points all on the same line cannot form a triangle; in other words, a triangle must have some area.)(A) 516 (B) 528 (C) 1632 (D) 3316 (E) 3344

Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]
03 Feb 2014, 12:04

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Number of ways to select 3 points out of 16 is 16C3 = 560. There is possibility that in some cases out of the 560 cases, the three points lie on the same line and therefore do not form a triangle. This eliminates options B,C and D

There are 4 columns and 4 rows make it a total of 8 linear possible arrangements of the points. Number of ways in which the points can be arranged along each row or column = 8*(4C3) = 8*4 = 32.

We are left with 560-32 =528 ways. Now there is also the possibility that the three points fall on a straight line if placed along the diagonal. Thus the number of ways is definitely less than 528, leaving option A. _________________

Paras.

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Re: In the above diagram, the 16 dots are in rows and columns, [#permalink]
03 Feb 2014, 23:56

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mikemcgarry wrote:

Attachment:

4x4 grid.JPG

In the above diagram, the 16 dots are in rows and columns, and are equally spaced in both the horizontal & vertical direction. How many triangles, of absolutely any shape, can be created from three dots in this diagram? Different orientations (reflections, rotations, etc.) and/or positions count as different triangles. (Notice that three points all on the same line cannot form a triangle; in other words, a triangle must have some area.)(A) 516 (B) 528 (C) 1632 (D) 3316 (E) 3344